To answer your question with a diagram.
Imagine the region of "=" as the part you are cutting.
I labeled where the upper and lower are.
I can answer the how the machine makes an educated guess about the position of the cutting head.
If you are using stepper motors, then the following is true:
The motor moves a know amount for every step it is given. When your computer give a step command to the motor, it contains 2 things: a direction (polarity of the step, if memory serves) and a pulse (the actual step).
The last robot I built had 1.8 degree/step motors in it. This means that every 200 steps you get a full 360 degree revolution.
The screw mechanism in the machine converts the rotation into a linear distance. The quick explanation is your screws have a TPI or threads per inch rating, for example, let's say 8 TPI. That means for every 8 revolutions your motor makes, the screw will move the nut 1" (ideally). In reality there is an equation that the controller uses to calculate the position, but it is subject to some assumptions, which breakdown if the machine skips steps.
Yes, I see no reason why not. There is usually some annoying paperwork to deal with though, if you want to be 100% compliant with the law (i.e. tax nonsense, business license, etc.). But while you are starting out, and making a little bit here and there, don't worry about it. Consult your city's administration if you want to know all of the ins and outs. I know San Diego has an office for small business assistance, but I don't know about where you live.
Hope that helps...